Solve for x log base 4 of x< 4

Solution:

Step 1:

Transform the given inequality into an equation: ${\log }_4(x)=4$.

Step 2:

Address the equation.

Step 2.1:

Express the logarithmic equation ${\log }_4(x)=4$ in its equivalent exponential form. For positive real numbers $x$ and $b$, with $b\ne 1$, ${\log }_b(x)=y$ corresponds to $b^y=x$. Thus, $4^4=x$.

Step 2.2:

Calculate the value of $x$.

Step 2.2.1:

Reformulate the equation to $x=4^4$.

Step 2.2.2:

Compute $4$ raised to the 4th power: $x=256$.

Step 3:

Identify the domain of the function ${\log }_4(x)-4$.

Step 3.1:

Ensure the argument inside the logarithm ${\log }_4(x)$ is greater than zero to determine the domain: $x>0$.

Step 3.2:

The domain is the set of all $x$ values that make the function valid: $(0,\mathrm{\infty })$.

Step 4:

Construct test intervals using the solution of the equation: $x<0$, $0<x<256$, $x>256$.

Step 5:

Select a test value from each interval and substitute it into the original inequality to verify which intervals satisfy the inequality.

Step 5.1:

Examine the interval $x<0$ with a test value to check if the inequality holds.

Step 5.1.1:

Pick a test value within the interval $x<0$: $x=-2$.

Step 5.1.2:

Substitute $x=-2$ into the original inequality: ${\log }_4(-2)<4$.

Step 5.1.3:

Determine the truth of the inequality.

Step 5.1.3.1:

The inequality is undefined as logarithms of negative numbers are not real.

Step 5.1.3.2:

The inequality does not hold as the left side is not defined.

Step 5.2:

Evaluate the interval $0<x<256$ with a test value to confirm the inequality.

Step 5.2.1:

Select a test value within $0<x<256$: $x=2$.

Step 5.2.2:

Insert $x=2$ into the original inequality: ${\log }_4(2)<4$.

Step 5.2.3:

The inequality is true as the left side ($0.5$) is less than the right side ($4$).

Step 5.3:

Assess the interval $x>256$ with a test value to validate the inequality.

Step 5.3.1:

Choose a test value greater than $256$: $x=258$.

Step 5.3.2:

Replace $x$ with $258$ in the inequality: ${\log }_4(258)<4$.

Step 5.3.3:

The inequality is false as the left side ($4.00561362$) is not less than $4$.

Step 5.4:

Contrast the intervals to ascertain which satisfy the original inequality: $x<0$ (False), $0<x<256$ (True), $x>256$ (False).

Step 6:

The solution is the union of all intervals where the inequality is true: $0<x<256$.

Step 7:

The solution can be represented in various formats:

Inequality Form: $0<x<256$

Interval Notation: $(0,256)$

Knowledge Notes:

The problem involves solving an inequality with a logarithmic function. Here are the relevant knowledge points:

1. Logarithms: The logarithm ${\log }_b(x)$ is the exponent to which the base $b$ must be raised to yield $x$. The base $b$ must be a positive real number and not equal to $1$.

2. Converting Logarithms to Exponential Form: The logarithmic equation ${\log }_b(x)=y$ can be rewritten in exponential form as $b^y=x$.

3. Domain of a Logarithmic Function: The domain of ${\log }_b(x)$ is $(0,\mathrm{\infty })$ because the logarithm of a negative number or zero is undefined in the real number system.

4. Solving Logarithmic Inequalities: To solve a logarithmic inequality, one can first solve the corresponding logarithmic equation to find critical values that partition the number line into intervals. Then, test values from each interval can be substituted into the original inequality to determine which intervals satisfy the inequality.

5. Interval Notation: Interval notation is a way of representing subsets of the real number line. An interval such as $(0,256)$ includes all numbers between $0$ and $256$, not including the endpoints themselves.

6. Exponents: Raising a number to a power, such as $4^4$, involves multiplying the number by itself the specified number of times. In this case, $4^4=4\times 4\times 4\times 4=256$.

7. Inequalities: An inequality like $0<x<256$ indicates that the value of $x$ is between $0$ and $256$, not including $0$ or $256$ themselves.